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\author{五六七}
\title{常微分方程第7周作业（6.2）}
%\date{2025年10月14日}

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%\abstract{第6章第2节：例1、例子2、例子3、例子4、习题1(1)、2(1)、3(1)。}

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\section{课堂练习(10.23)}
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\begin{enumerate}
\item  %例子6.2\#2A
设 $A=\begin{bmatrix} 2 & 1 \\  0 & 2 \end{bmatrix}$, 求 $\exp(A)$.

\item  %例子6.2\#4A
求常系数齐次线性微分方程组 $ \frac{d}{dt}\begin{bmatrix} x \\ y \end{bmatrix}=\begin{bmatrix} -2 & 2 \\  -10 & 7 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}$ 的通解。

\item  %习题6.2\#2(1)A
求常系数非齐次线性微分方程组 $\frac{d}{dt}\begin{bmatrix} x \\ y \end{bmatrix}=\begin{bmatrix} -2 & 2 \\ -10 & 7 \end{bmatrix}\begin{bmatrix} x \\ y \end{bmatrix} + \begin{bmatrix} 1 \\ 2 \end{bmatrix}$ 的通解。

\item  %例子6.3\#3A
求高阶线性微分方程 $y''+y'-6y=x$ 的通解。

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\section{课后作业（10.21-10.27）}

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\item  例子6.2\#1
设 $A=\begin{bmatrix} a_1 & 0 \\ 0 & a_2 \end{bmatrix}$, 求 $e^A$.

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\item  例子6.2\#2
设 $A=\begin{bmatrix} 1 & 1 \\  0 & 1 \end{bmatrix}$, 求 $e^A$.

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\item  例子6.2\#3
求微分方程组 $\frac{d\vec{y}}{dx}=A\vec{y}$ 的通解，其中矩阵 
$$A=\begin{bmatrix} 5 & -28 & -18 \\  -1 & 5 & 3 \\ 3 & -16 & -10 \end{bmatrix}.$$  
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\item  例子6.2\#4
求微分方程组 $ \frac{d\vec{y}}{dx}=\begin{bmatrix} 1 & 1 \\  -1 & 1 \end{bmatrix} \vec{y}$ 的通解。
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\item  习题6.2\#1(1)
求常系数齐次线性微分方程组 $\frac{d \vec{y}}{dx}=A\vec{y}$ 的通解，其中矩阵 $A$ 为
$$
A = \begin{bmatrix} 3 & 4 \\ 5 & 2 \end{bmatrix}.
$$ 

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\item  习题6.2\#2(1)
求常系数非齐次线性微分方程组 $\frac{d \vec{y}}{dx}=A\vec{y} + \vec{f}(x)$ 的通解，其中 
$$A = \begin{bmatrix} 2 & 1 \\ 0 & 2 \end{bmatrix},\,\, \vec{f}(x) = \begin{bmatrix} 1 \\ 0 \end{bmatrix}. $$
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\item  习题6.2\#3(1)
求微分方程组 $\frac{d \vec{y}}{dx}=A\vec{y} + \vec{f}(x)$ 满足初值条件 $\vec{y}(0)=\vec{\eta}$ 的解，其中
$$A = \begin{bmatrix} -5 & -1 \\ 1 & -3 \end{bmatrix},\,\, 
\vec{f}(x) = \begin{bmatrix} e^x \\ e^{2x} \end{bmatrix},\,\, 
\vec{\eta} = \begin{bmatrix} 1 \\ 0 \end{bmatrix}.
$$ 

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